Dimension Properties Of Sample Paths Of N-parameter Operator Stable Lévy Processes

Operator stable Levy processes are Levy processes and more extensive than stable processes. Its corresponding probability distribution is called operator stable distribution. Together, they are used to depict heavy tailed random vectors with diverse tail behaviors, and therefore carry significant meaning in the field of natural science and financial insurance. Becket-kern et al. researched the Hausdorff dimen-sion for the range of operator stable Levy processes. Yet no discussion has been made concerning multi-parameter situation. In this paper, we focus on dimensional properties concerning sample paths of N-parameter operator stable Levy processes. We obtained the following results:LetⅩ＝{x(t), t∈RN+} be a N-parameter operator stable Levy process in Rd with exponent B, where B is a d×d invertible matrix.α1＞α2＞...＞αp are the reciprocal of the real parts of p different eigenvalues of B, and d1,...,dp are dimensions of corresponding subspaces respectively.(One) The Hausdorff uniform dimension upper bounds of range and graph sets ofⅩ.(Two) The Hausdorff dimension lower bounds of range and graph sets ofⅩ.(Three) The Hausdorff dimension of range and graph sets ofⅩ. The above results show that the Hausdorff dimension of the range and graph sets of N-parameter operator stable Lévy process X is completely determined by the real parts of B's eigenvalues. Besides, paper [1] and one of the main results of paper [2] (Theorem 3.8) are two special cases of our results.